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Mixed finite element approximation for a coupled petroleum reservoir model

Published online by Cambridge University Press:  15 April 2005

Mohamed Amara ,
Daniela Capatina-Papaghiuc ,
Bertrand Denel  and
Peppino Terpolilli
Mohamed Amara
Affiliation:
Université de Pau, L.M.A. CNRS-FRE 2570, Avenue de l'Université, 64000 Pau, France. mohamed.amara@univ-pau.fr; daniela.capatina-papaghiuc@univ-pau.fr; bertrand.denel@univ-pau.fr
Daniela Capatina-Papaghiuc
Affiliation:
Université de Pau, L.M.A. CNRS-FRE 2570, Avenue de l'Université, 64000 Pau, France. mohamed.amara@univ-pau.fr; daniela.capatina-papaghiuc@univ-pau.fr; bertrand.denel@univ-pau.fr
Bertrand Denel
Affiliation:
Université de Pau, L.M.A. CNRS-FRE 2570, Avenue de l'Université, 64000 Pau, France. mohamed.amara@univ-pau.fr; daniela.capatina-papaghiuc@univ-pau.fr; bertrand.denel@univ-pau.fr
Peppino Terpolilli
Affiliation:
Total, CST Jean Feger, Avenue Larribau, 64018 Pau Cedex, France. peppino.terpolilli@total.com
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Abstract

In this paper, we are interested in the modelling and the finite element approximation of a petroleum reservoir, in axisymmetric form. The flow in the porous medium is governed by the Darcy-Forchheimer equation coupled with a rather exhaustive energy equation. The semi-discretized problem is put under a mixed variational formulation, whose approximation is achieved by means of conservative Raviart-Thomas elements for the fluxes and of piecewise constant elements for the pressure and the temperature. The discrete problem thus obtained is well-posed and a posteriori error estimates are also established. Numerical tests are presented validating the developed code.

Keywords

Petroleum reservoirthermometricsporous mediummixed finite elementsa posteriori estimators.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 39 , Issue 2 , March 2005 , pp. 349 - 376
Copyright
© EDP Sciences, SMAI, 2005

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References

C. Abchir, Modélisation des écoulements dans les réservoirs souterrains avec prise en compte des interactions puits/réservoir. Thèse de doctorat, Université de Saint-Etienne (1992).
M. Amara, D. Capatina, B. Denel and P. Terpolilli, Modelling, analysis and numerical approximation of flow with heat transfer in a petroleum reservoir, Preprint No. 0415, Université de Pau (2004) (http://lma.univ-pau.fr/publis/publis.php).
G. Bourdarot, Well testing: Interpretation methods. Editions Technip, Paris (1998).
S. Brenner and R. Scott, The mathematical theory of Finite Element Methods. Springer Verlag, New York (1994).
F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer Verlag, New York (1991).
Chavent, G. and Roberts, J.E., A unified physical presentation of mixed, mixed-hybrid finite elements and standard finite difference approximations for the determination of velocities in waterflow problems. Adv. Water Resources 14 (1991) 329–348. CrossRef
P.G. Ciarlet, The finite element method for elliptic problems error analysis. North Holland, Amsterdam (1978).
Ewing, R.E., Wang, J. and Weekes, S.L., On the simulation of multicomponent gas flow in porous media. Appl. Numer. Math. 31 (1999) 405–427. CrossRef
P. Grisvard, Elliptic problems on non-smooth domains. Pitman, Boston (1985).
F. Maubeuge, M. Didek, E. Arquis, O. Bertrand and J.-P. Caltagirone, Mother: A model for interpreting thermometrics. SPE 28588 (1994).
Peng, D.Y. and Robinson, D.B., A new two-constant equation of state. Ind. Eng. Chem. Fundam. 15 (1976) 59–64. CrossRef
J.E. Roberts and J.-M. Thomas, Mixed and Hybrid Methods, in Handbook of Numerical Analysis Vol. II. North Holland, Amsterdam (1991) 523–639.
Verfürth, R. and Braess, D., A posteriori error estimator for the Raviart-Thomas element. SIAM J. Numer. Anal. 33 (1996) 2431–2444.